Definitions and introduction. Let Ц = {Ui|i ∈ I} be a system of subsets of a normal topological space R; i.e. a mapping from the index set I into the set of all subsets of R. The order of a point x is the number of distinct member sets of Ц which contain x, and is denoted by x: Ц; the sets Ui are here considered distinct if they have distinct indices. Thus x: Ц is the number of indices i for which x ∈ Ui; ν(Ц) = max {x: Ц | x ∈ R} is called the order of the system Ц. If every point has an (open) neighbourhood meeting only finitely many members of Ц, then Ц is said to be locally finite.